Question: Find the gradient of $f(x, y) = x^2 + y^2$. $\nabla f = ($ $,$ $)$
Explanation: The gradient of a scalar field is all its partial derivatives put together into a vector. For a 2D scalar field, this looks like $\nabla f = (f_x, f_y)$. Let's find $f_x$ and $f_y$. $\begin{aligned} f_x &= \dfrac{\partial}{\partial x} \left[ x^2 + y^2 \right] \\ \\ &= 2x \\ \\ f_y &= \dfrac{\partial}{\partial y} \left[ x^2 + y^2 \right] \\ \\ &= 2y \end{aligned}$ The gradient of $f$ is $\nabla f = (2x, 2y)$.